### By Mark Dawes, April 2024

The mid-morning show on BBC Radio 2 used to feature a pop music quiz called *PopMaster*. Contestants scored 3 points for a correct answer to a question and 6 points for a correct bonus question. Each player was asked 10 questions, of which three were bonus questions. This gave a maximum score of 39. I never heard part-marks being awarded, so all scores were a multiple of 3. This seemed rather … unusual (why not divide everything by 3?).

Since the originator (and owner) of *PopMaster* moved on and took the quiz with him, Vernon Kay has used a new quiz called *Ten to the Top*. The scoring system is even more bizarre, but mathematically much more interesting!

Here are the basic rules (we’ll start by ignoring the joker):

Here are some questions that a class might enjoy exploring.

Q1) If this is your scorecard, what is your total?

Q2) If you get all 10 answers correct (according to these, partial rules), what would you score?

Q3) If you get question 7 wrong but all the rest are correct, what do you score?

Q4) If you get nine of the questions right, what is the biggest score you can get? What is the smallest score? Explain.

Q5) If you get eight of the questions right, what is the biggest score you can get? What is the smallest score?

Q6) Is it possible for a player who gets 5 correct answers to beat someone who gets 7 correct answers?

Q7) Show how a player who gets 4 correct answers could draw against one who gets 7 correct answers.

I mentioned that there is one additional rule:

If you don’t play it earlier in the quiz, then it is automatically played on question 10.

Clearly, it makes sense to play the joker as late as possible so you can double up a higher number, but you don’t want to risk getting one wrong and resetting to start again at 1!

Q8) What is the actual highest score you can get in the game, including considering the joker?

Q9) Using a joker, show how it is possible for someone getting 6 correct to beat someone getting 9 correct.

Q10) Using a joker, it is possible to score almost every number up to the highest score. I think there is only one total it is impossible to get. Can you find it? [Hard question!]

Q11) Pick a total and try to make it using the rules of the game (and involving the joker). Can you make your chosen number in more than one way?

##### Here are some answers:

A1) If this is your scorecard, what is your total?

This gives a total of 17

A2) If you get all 10 answers correct (according to these, partial rules), what would you score?

1+2+3+4+5+6+7+8+9+10 = 55

A3) If you get question 7 wrong but all the rest are correct, what do you score?

1+2+3+4+5+6+0+1+2+3 = 27

A4) If you get nine of the questions right, what is the biggest score you can get? What is the smallest score? Explain.

Best case is if you get question 1 or question 10 wrong. That gives you 1+2+3+4+5+6+7+8+9 = 45

Worst is if you get question 5 or 6 wrong, because that interrupts your streak. This gives 1+2+3+4+5+0+1+2+3+4 = 25

A5) If you get eight of the questions right, what is the biggest score you can get? What is the smallest score?

Best case is to get first two or last two wrong: 1+2+3+4+5+6+7+8 = 36

Worst is to make the streaks as short as possible. Here is one way: 123X123X12, which totals 15.

A6) Is it possible for a player who gets 5 correct answers to beat someone who gets 7 correct answers?

Yes! Best score for five correct is 1+2+3+4+5=15

Worst score for 7 is 12X12X12X1, which gives 10.

A7) Show how a player who gets 4 correct answers could draw against one who gets 7 correct answers.

1234XXXXXX gives 10 , which is the same as the worst for 7 (see A6).

A8) What is the actual highest score you can get in the game, including considering the joker?

Getting all the questions right and playing the joker on question 10 would double the score of 10. This gives a total of 65.

A9) Using a joker, show how it is possible for someone getting 6 correct to beat someone getting 9 correct.

If you get the first 6 questions correct and play the joker on question 6 you would score 1+2+3+4+5+12 = 27

We have already seen that if you get only question 6 wrong (and play your joker on that question) then you would get a total of 25.

A10) Using a joker, it is possible to score almost every number up to the highest score. I think there is only one total it is impossible to get. Can you find it? [Hard question!]

I think 55 is the only impossible score from zero up to 65.

Here are the triangular numbers:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55.

(For example, 10 is the fourth triangular number because 1+2+3+4 = 10

And 21 is the sixth triangular number because 1+2+3+4+5+6 = 21.)

We can get 21 by getting the first 6 questions right, the rest wrong, and playing the joker on a wrong question.

We can get 22 by getting the first 6 right, the rest wrong, and playing the joker on question 1.

We can get 23 by getting the first 6 right, the rest wrong, and playing the joker on question 2.

We can get 24 by getting the first 6 right, the rest wrong, and playing the joker on question 3.

etc, up to :

We can get 27 by getting the first 6 right, the rest wrong, and playing the joker on question 6.

There are other ways to get each of these totals, but it is rather neat to do it like this, because it gives a way to show how to get the totals from 28 up 35 (get the first 7 right, the rest wrong and use the joker on different questions).

And this then extends to the other numbers too.

Can pupils see why the only number that doesn’t work is 55? This is because to score 55 you must get all 10 questions correct (getting nine right and doubling the 9 only gives 54), but you have to play the joker at some point, so one of them will be doubled, which will give a total bigger than 55.

A11) Pick a total and try to make it using the rules of the game (and involving the joker). Can you make your chosen number in more than one way?

The method shown in A10 gives one way to do this. Fairly small scores (say 10) have lots of ways.

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